# What is factor sensitivity for a portfolio

Hi, everyone

I am taking the May Part 1 exam and I am using Kaplan notes for studying. There is a Practice question at the end of foundations of Risk Management and I am struggling to understand,

The question is to find out a volatility of a investment to a two-factor model. It gives out a factor covariance matrix to two factors, say Factor A and Factor B.

The covariance for A to A is 0.0245

The covariance for B to B is 0.01250

The covariance for A to B or B to A is 0.00791.

It also gives out the factor sensitivity for A is 0.75 and factor sensitivity for B is 0.20.

The answer for this question is Variance for this investment, which works out by this formula

Factor Sensitivity A ^2 x Cov (A, A) + Factor Sensitivity B ^2 * x Cov (B, B) + 2 x Factory Sensitivity A x Factory Sensitivity B x Cov (A, B)

= (0.75)^2 x (0.245) + (0.20)^2 x (0.0125) + 2 x (0.75) x (0.20) x (0.00791)

The answer also calls the Factor Sensitivity A is Beta A.

The only formula I can link this answer to is the Topic 2’s

Variance = W(A)^2 x Var A + W(B)^2 x Var B + 2xW(A)xW(B)xCov(A,B)

But this formula uses Weight, rather than Beta. My question is where the formula in answer comes from and also why Factor Sensitivity is considered as Beta? What is Factor Senstivity?

Many many thanks,

Kai

actually, the quesiton can be found on this link,

http://www.garp.org/media/613028/frm%20practice%20exam052711.pdf

question 7 on page 52. Thx

Factors in APT could be seen as securities. Then security weights are equivalent to betas (factor sensitivities). The quations are exactly the same.

thanks. Ara1523